Pythagoras’ Theorem
Pythagoras’ Theorem states:
The length of the hypotenuse squared equals the sum of
the squares of the other two sides.
Pythagoras
The Learning Centre
When we represent the other two sides by a and b we could write
Pythagoras’ theorem in symbols.
h 2 = a2 + b 2 .
To be able to use this formula, all the lengths of the sides must be
measured in the same units.
Exercises:
1. What is the length of the hypotenuse of the following triangle?
Solutions:
1. From Pythagoras’ Theorem,
h 2 = a2 + b 2 .
In this situation, a = 2 , b = 9 , h = ? .
It would not matter which side we let be a and which side we
let be b. So,
2. Find the length of the unknown side.
h 2 = 22 + 92 ,
h2 = 85
√,
so h =
85 ≈ 9.22 .
Note the negative square root in this
case has no meaning so we disregard
it.
The length of the hypotenuse is approximately 9.22 cm.
Solutions (cont)
More exercises
2. Pythagoras’ Theorem states
h 2 = a2 + b 2 .
1. For the following right angled triangles, find the unknown
length (to two decimal places if necessary).
Here, a = 3, b = x, h = 6.5 . So,
(6.5)2
42.25
42.25 − 9
x2
so x
=
=
=
=
=
32 + x 2 ,
9 + x2 ,
x2 ,
33.25
,
√
33.25 ≈ 5.8 .
Take 9 from both sides.
Again we disregard the
negative square root.
So the length of the unknown side is approximately 5.8 m.
Answers
Answers (cont)
1.
x2
= 4 2 + 32 ,
x2
= 16 + 9 ,
x
2
x
= 25 ,
√
=
25 ,
x
= 5.
Thus the unknown length is 5 cm.
2.
(10.6)2
= (3.7)2 + x 2 ,
112.36 = 13.69 + x 2 ,
x2
= 112.36 − 13.69 ,
x2
x
= 98.67 ,
√
=
98.67 ,
x
≈ 9.93 .
Thus the unknown length is approximately 9.93 m.
3.
962 = 572 + x 2 ,
9 216 = 3 249 + x 2 ,
x 2 = 9 216 − 3 249 ,
x 2 = 5 967 ,
√
x =
5 967 ,
x
≈ 77.25 .
Thus the unknown length is approximately 77.25 mm.